Fixing the UPCC case of Split-or-Johnson

In November 2015 I announced the result that Graph Isomorphism can be tested in quasipolynomial time. The first version of the full paper was posted on arXiv in December 2015, and an updated version in January 2016 (arXiv:1512.03547). Below we refer to Version 2 (Jan 2016) of that paper as [Q2]. On January 1, 2017, Harald Helfgott pointed out an error in the timing analysis of one of the combinatorial Divide-and-Conquer tools, the “Split-or-Johnson” routine (SoJ). The error invalidated the quasipolynomial claim. A revised analysis of a slightly modified algorithm (adjusting a threshold parameter for optimal analysis) gave a running time bound of exp exp(Õ( √ log n)) where the Õ hides a poly-loglog term. While still subexponential (less than exp(n ) for all positive ) and thus considerably stronger than the previously known moderately exponential bound of exp(Õ( √ n)) (Luks, 1983), this bound was not quasipolynomial. I announced the revised bound on my home page on January 4, 2017. Three days later I found a simple fix and announced this on January 9, 2017 both on the website and in a lecture I gave that day at Georgia Tech. So the quasipolynomial claim has been restored. Meanwhile I have been working on a major revision of the arXiv posting, taking into account a large number of helpful comments from colleagues. The intention of the update is to improve the presentation, give more detailed proofs and analysis, and also to fix another, less dramatic, error that was pointed out to me by Jin-Yi Cai in Spring 2016; that error occurred in the “Design Lemma” algorithm (DL). My analysis of the modified algorithm for the DL was additionally verified by Gábor Tardos and by Harald Helfgott, both of whom also contributed many comments on how to improve the presentation. Since the completion of the revision of the paper is still months away, and given the explosive interest in the recent (SoJ) update, I decided to post this update in advance of the revision of the full paper. I will also post the DL update in the near future. This note requires some familiarity with [Q2]. The error was in Case 8(iii) (“uniprimitive (UPCC) case”) of the “Bipartite Split-orJohnson” algorithm (Sec. 7.5 in [Q2]) to which the proposed solution (Sec. 7.9 in [Q2]) involved a recursive call to the “UPCC Split-or-Johnson” algorithm. This recursive call caused an impermissible blow-up in the running time. The present note shows how to replace this “malignant” recursive call by a “benign” one that leads to the same type of recurrence that occurs in many other places in the paper and leads to a quasipolynomial solution. This update also subsumes Case 8(iv) (“Johnson case”) of the “Bipartite Split-or-Johnson” algorithm; consequently, that case (Secs. 7.10 and 7.11 in [Q2]) is now eliminated.